Germany | A nice Math Olympiad Logarithmic Equation | Exponential Simplification Problem | Find x=?

[@YAWTon]

Obviously there is only one real solution (from monotonicity of LHS). The solution x satisfies 3^log(x)=4, which is easy to solve for x (x=4^(1/log(3)), assuming natural log).

[@juanalfaro7522]

Let y=3^log (x), then y^3 + y = 68. Clearly y=4 is the only solution because is a solution and y^3 + y is an increasing function (f'(x) = 3x^2 + 1 > 0). Thus log (x) = log (4)/log (3) -> x = 10^(log(4)/log(3)) = 18.2751

[@baselinesweb]

Nice job

[@superacademy247]

Thanks!

[@UKPEINDANIELU.]

Fantastic

[@superacademy247]

Thank you so much 😀

[@ludmilak9396]

А почему нужно было основание 10? Разве нельзя использовать любое иное основание?

[@superacademy247]

Base 10 is a common base. And I wanted the result to look beautiful. You can choose any base if you don't care about fancy solution. And most importantly, base 10 enables us to get rid of the logarithm on the LHS, a clear way of obtaining the value of x.